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arXiv · 1502.03704

Improved bounds for arithmetic progressions in product sets

Dmitry Zhelezov
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Abstract

Let $B$ be a set of natural numbers of size $n$. We prove that the length of the longest arithmetic progression contained in the product set $B.B = \{bb'| \, b, b' \in B\}$ cannot be greater than $O(n \log n)$ which matches the lower bound provided in an earlier paper up to a multiplicative constant. For sets of complex numbers we improve the bound to $O_\epsilon(n^{1 + \epsilon})$ for arbitrary $\epsilon > 0$ assuming the GRH.

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Dmitry Zhelezov. 2015-02-12. Improved bounds for arithmetic progressions in product sets. https://arxiv.org/abs/1502.03704

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